In

composition of relations
In the of s, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the , the composition of relations is called relative multiplication, and its result is called a relative produc ...

using

Composition of Functions

by Bruce Atwood, the

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, function composition is an operation that takes two functions
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

and and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and are composed to yield a function that maps in to in .
Intuitively, if is a function of , and is a function of , then is a function of . The resulting ''composite'' function is denoted , defined by for all in .
The notation is read as " circle ", " round ", " about ", " composed with ", " after ", " following ", " of ", " then ", or " on ", or "the composition of and ". Intuitively, composing functions is a chaining process in which the output of function feeds the input of function .
The composition of functions is a special case of the composition of relations
In the of s, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the , the composition of relations is called relative multiplication, and its result is called a relative produc ...

, sometimes also denoted by $\backslash circ$. As a result, all properties of composition of relations are true of composition of functions, though the composition of functions has some additional properties.
Composition of functions is different from multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

of functions, and has quite different properties; in particular, composition of functions is not commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

.
Examples

* Composition of functions on a finite set: If , and , then , as shown in the figure. * Composition of functions on aninfinite set
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...

: If (where is the set of all real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s) is given by and is given by , then:
:, and
:.
* If an airplane's altitude at time is , and the air pressure at altitude is , then is the pressure around the plane at time .
Properties

The composition of functions is alwaysassociative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

—a property inherited from the composition of relations
In the of s, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the , the composition of relations is called relative multiplication, and its result is called a relative produc ...

. That is, if , , and are composable, then . Since the parentheses do not change the result, they are generally omitted.
In a strict sense, the composition is only meaningful if the codomain of equals the domain of ; in a wider sense, it is sufficient that the former be a subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of the latter.
Moreover, it is often convenient to tacitly restrict the domain of , such that produces only values in the domain of . For example, the composition of the functions defined by and defined by $g(x)\; =\; \backslash sqrt\; x$ can be defined on the interval .
The functions and are said to commute with each other if . Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, only when . The picture shows another example.
The composition of one-to-one
One-to-one or one to one may refer to:
Mathematics and communication
*One-to-one function, also called an injective function
*One-to-one correspondence, also called a bijective function
*One-to-one (communication), the act of an individual commun ...

(injective) functions is always one-to-one. Similarly, the composition of onto
In , a surjective function (also known as surjection, or onto function) is a that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's is the of one element of its ...

(surjective) functions is always onto. It follows that the composition of two bijection
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s is also a bijection. The inverse function
In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...

of a composition (assumed invertible) has the property that .
Derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s of compositions involving differentiable functions can be found using the chain rule. Higher derivative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s of such functions are given by Faà di Bruno's formula.
Composition monoids

Suppose one has two (or more) functions having the same domain and codomain; these are often called ''transformations
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Transf ...

''. Then one can form chains of transformations composed together, such as . Such chains have the algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of a monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

, called a ''transformation monoidIn algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...

'' or (much more seldom) a ''composition monoid''. In general, transformation monoids can have remarkably complicated structure. One particular notable example is the de Rham curve#Redirect DE {{Redirect category shell, 1=
{{Redirect from other capitalisation
{{Redirect from ambiguous term
...

. The set of ''all'' functions is called the full transformation semigroupIn algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...

or ''symmetric semigroup'' on . (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions.)
If the transformations are bijective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

(and thus invertible), then the set of all possible combinations of these functions forms a transformation group
In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...

; and one says that the group is generated by these functions. A fundamental result in group theory, Cayley's theorem
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group (mathematics), group ''G'' is group isomorphism, isomorphic to a subgroup of the symmetric group acting on ''G''. This can be understood as an example of ...

, essentially says that any group is in fact just a subgroup of a permutation group (up to isomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

).
The set of all bijective functions (called permutation
In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

s) forms a group with respect to function composition. This is the symmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

, also sometimes called the ''composition group''.
In the symmetric semigroup (of all transformations) one also finds a weaker, non-unique notion of inverse (called a pseudoinverse) because the symmetric semigroup is a regular semigroupIn mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...

.
Functional powers

If , then may compose with itself; this is sometimes denoted as . That is: : : : More generally, for anynatural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

, the th functional power
Power typically refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, p ...

can be defined inductively by , a notation introduced by Hans Heinrich Bürmann and John Frederick William Herschel
Sir John Frederick William Herschel, 1st Baronet (; 7 March 1792 – 11 May 1871) was an English polymath
A polymath ( el, πολυμαθής, ', "having learned much"; Latin
Latin (, or , ) is a classical language belonging to th ...

. Repeated composition of such a function with itself is called iterated function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

.
* By convention, is defined as the identity map on 's domain, .
* If even and admits an inverse function
In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...

, negative functional powers are defined for as the negated
In logic, negation, also called the logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted i ...

power of the inverse function: .
Note: If takes its values in a ring (in particular for real or complex-valued ), there is a risk of confusion, as could also stand for the -fold product of , e.g. . For trigonometric functions, usually the latter is meant, at least for positive exponents. For example, in trigonometry
Trigonometry (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...

, this superscript notation represents standard exponentiation
Exponentiation is a mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...

when used with trigonometric functions
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

:
.
However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., .
In some cases, when, for a given function , the equation has a unique solution , that function can be defined as the functional square root
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of , then written as .
More generally, when has a unique solution for some natural number , then can be defined as .
Under additional restrictions, this idea can be generalized so that the iteration count becomes a continuous parameter; in this case, such a system is called a flow
Flow may refer to:
Science and technology
* Flow (fluid) or fluid dynamics, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on ...

, specified through solutions of Schröder's equation (1841–1902) in 1870 formulated his eponymous equation.
Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable
Dependent and independent variables are Variable and attribute (research), varia ...

. Iterated functions and flows occur naturally in the study of fractals
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

and dynamical systems
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. Examples include the mathematical models that describe the ...

.
To avoid ambiguity, some mathematicians choose to use to denote the compositional meaning, writing for the -th iterate of the function , as in, for example, meaning . For the same purpose, was used by Benjamin Peirce
Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

whereas Alfred Pringsheim
Alfred Pringsheim (2 September 1850 – 25 June 1941) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quan ...

and Jules Molk
Jules Molk (8 December 1857 in Strasbourg, France – 7 May 1914 in Nancy) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of ...

suggested instead.
Alternative notations

Many mathematicians, particularly ingroup theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, omit the composition symbol, writing for .
In the mid-20th century, some mathematicians decided that writing "" to mean "first apply , then apply " was too confusing and decided to change notations. They write "" for "" and "" for "". This can be more natural and seem simpler than writing functions on the left in some areas – in linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mat ...

, for instance, when is a row vector
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and th ...

and and denote matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...

and the composition is by matrix multiplication
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. This alternative notation is called postfix notation
Reverse Polish notation (RPN), also known as Polish postfix notation or simply postfix notation, is a mathematical notation in which operators ''follow'' their operands, in contrast to Polish notation (PN), in which operators ''precede'' their ...

. The order is important because function composition is not necessarily commutative (e.g. matrix multiplication). Successive transformations applying and composing to the right agrees with the left-to-right reading sequence.
Mathematicians who use postfix notation may write "", meaning first apply and then apply , in keeping with the order the symbols occur in postfix notation, thus making the notation "" ambiguous. Computer scientists may write "" for this, thereby disambiguating the order of composition. To distinguish the left composition operator from a text semicolon, in the Z notation
The Z notation is a formal specification language used for describing and modelling computing systems. It is targeted at the clear specification of computer program
A computer program is a collection of instructions that can be executed by ...

the ⨾ character is used for left relation composition
Relation or relations may refer to:
General uses
* International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship
The concept of interpersonal relationship involves social ...

. Since all functions are binary relations
Binary may refer to:
Science and technology
Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...

, it is correct to use the semicolon for function composition as well (see the article on composition of relations
In the of s, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the , the composition of relations is called relative multiplication, and its result is called a relative produc ...

for further details on this notation).
Composition operator

Given a function , the composition operator is defined as that operator which maps functions to functions as ::$C\_g\; f\; =\; f\; \backslash circ\; g.$ Composition operators are studied in the field ofoperator theoryIn mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or ...

.
In programming languages

Function composition appears in one form or another in numerousprogramming language
A programming language is a formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

s.
Multivariate functions

Partial composition is possible formultivariate function
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (math ...

s. The function resulting when some argument of the function is replaced by the function is called a composition of and in some computer engineering contexts, and is denoted
:$f,\; \_\; =\; f\; (x\_1,\; \backslash ldots,\; x\_,\; g(x\_1,\; x\_2,\; \backslash ldots,\; x\_n),\; x\_,\; \backslash ldots,\; x\_n).$
When is a simple constant , composition degenerates into a (partial) valuation, whose result is also known as restriction or ''co-factor''.
:$f,\; \_\; =\; f\; (x\_1,\; \backslash ldots,\; x\_,\; b,\; x\_,\; \backslash ldots,\; x\_n).$
In general, the composition of multivariate functions may involve several other functions as arguments, as in the definition of primitive recursive function
In computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degree ...

. Given , a -ary function, and -ary functions , the composition of with , is the -ary function
:$h(x\_1,\backslash ldots,x\_m)\; =\; f(g\_1(x\_1,\backslash ldots,x\_m),\backslash ldots,g\_n(x\_1,\backslash ldots,x\_m))$.
This is sometimes called the generalized composite or superposition of ''f'' with . The partial composition in only one argument mentioned previously can be instantiated from this more general scheme by setting all argument functions except one to be suitably chosen projection functions. Here can be seen as a single vector/tuple
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

-valued function in this generalized scheme, in which case this is precisely the standard definition of function composition.
A set of finitary operations on some base set ''X'' is called a clone
Clone or Clones or Cloning or The Clone may refer to:
Places
* Clones, County Fermanagh
* Clones, County Monaghan, a town in Ireland
Biology
* Clone (B-cell), a lymphocyte clone, the massive presence of which may indicate a pathological conditio ...

if it contains all projections and is closed under generalized composition. Note that a clone generally contains operations of various arities. The notion of commutation also finds an interesting generalization in the multivariate case; a function ''f'' of arity ''n'' is said to commute with a function ''g'' of arity ''m'' if ''f'' is a homomorphism
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

preserving ''g'', and vice versa i.e.:
:$f(g(a\_,\backslash ldots,a\_),\backslash ldots,g(a\_,\backslash ldots,a\_))\; =\; g(f(a\_,\backslash ldots,a\_),\backslash ldots,f(a\_,\backslash ldots,a\_))$.
A unary operation always commutes with itself, but this is not necessarily the case for a binary (or higher arity) operation. A binary (or higher arity) operation that commutes with itself is called medial or entropic.
Generalizations

Composition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The term composition means "putting togethe ...

can be generalized to arbitrary binary relation
Binary may refer to:
Science and technology
Mathematics
* Binary number
In mathematics and digital electronics
Digital electronics is a field of electronics
The field of electronics is a branch of physics and electrical engineeri ...

s.
If and are two binary relations, then their composition is the relation defined as .
Considering a function as a special case of a binary relation (namely functional relation
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
In architecture, functionalism is the principle that buildings should be designed based solely on their purpose and function.
This principle is a matter of co ...

s), function composition satisfies the definition for relation composition. A small circle has been used for the infix notation of composition of relations, as well as functions. When used to represent composition of functions $(g\; \backslash circ\; f)(x)\; \backslash \; =\; \backslash \; g(f(x))$ however, the text sequence is reversed to illustrate the different operation sequences accordingly.
The composition is defined in the same way for partial function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s and Cayley's theorem has its analogue called the Wagner–Preston theoremIn group (mathematics), group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that ''x = xyx'' and ''y = yxy'', ...

.
The category of sets In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

with functions as morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s is the prototypical category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

. The axioms of a category are in fact inspired from the properties (and also the definition) of function composition. The structures given by composition are axiomatized and generalized in category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

with the concept of morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

as the category-theoretical replacement of functions. The reversed order of composition in the formula applies for converse relation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s, and thus in group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

. These structures form dagger categories.
Typography

The composition symbol is encoded as ; see theDegree symbol
The degree symbol or degree sign, , is a typographical symbol
This is a list of the most common typographical symbols and punctuation marks used in western European languages. For a far more comprehensive list of symbols and signs, see List of U ...

article for similar-appearing Unicode characters. In TeX
TeX (, see below), stylized within the system as TeX, is a typesetting system which was designed and mostly written by Donald Knuth and released in 1978. TeX is a popular means of typesetting complex mathematical formulae; it has been noted ...

, it is written `\circ`

.
See also

*Cobweb plot
upright=1.2, An animated cobweb diagram of the logistic map y = r x (1-x), showing chaos theory, chaotic behaviour for most values of r > 3.57.
A cobweb plot, or Verhulst diagram is a visual tool used in the dynamical systems field of mathematics ...

– a graphical technique for functional composition
* Combinatory logic
Combinatory logic is a notation to eliminate the need for Quantifier (logic), quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoreti ...

* Composition ring, a formal axiomatization of the composition operation
* Flow (mathematics)
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

* Function composition (computer science)
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of Algori ...

* Function of random variable, distribution of a function of a random variable
* Functional decomposition
In mathematics, functional decomposition is the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts by function composition.
...

* Functional square root
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

* Higher-order function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

* Infinite compositions of analytic functions In mathematics, infinite composition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The te ...

* Iterated function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

* Lambda calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system
A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, ar ...

Notes

References

External links

* {{springer, title=Composite function, id=p/c024260 *Composition of Functions

by Bruce Atwood, the

Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hoste ...

, 2007.
Functions and mappings
Basic concepts in set theory{{Commons
This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed.
Mathematical concepts ...

Binary operations